In the case of a polymer in an open space \(\mathbb {R}^3\) , the standard Kratky-Porod formula for polymers is reproduced in three dimensions [1]}, while when the polymer is confined to a space bounded region \(\mathcal {D}\subset \mathbb {R}^{3}\)  the conformational states shows the existence of a critical persistence length \(\ell _{p}^{*}\)  such that for all values of \(\ell _{p}>\ell _{p}^{*}\)  the mean square distance from end-to-end exhibits an oscillating behavior, while for \(\ell _{p} <\ell _{p}^{*}\)  it exhibits a monotonic behavior in both cases of a cubic region and a spherical region . Furthermore, for each value of \(\ell _{p}\) , the function converges to the twice of the mean-square end position \(\sigma ^{2}\left({\bf R}\right)\) , that is, twice the variance of \({\bf R}^2\)  with respect to the volume of the domain.
The critical persistence length, therefore, distinguishes two conformational behaviors of the semiflexible polymer in the bound domain. On one hand, polymers with persistence length below the critical value have a conformation similar to a Brownian random path. On the other hand, polymers with persistence length above the critical value adopt smooth conformations.
In addition, it is highlighted that the mean square end-to-end distance exhibits the same mathematical form for the discussed cases along with the manuscript (see Eq.(31) and Eq. (38)), and with the results reported for a polymer enclosed to a square box and rolling up a spherical surface [2]}, [3]}.
Nevertheless, the values-difference of saturation and the critical persistence length reflects the particular geometric nature of the compact domain, including the dimensionality of the space.
Note the particular mathematical expression in our work is due to the probability density function of the polymer's ends which is governed by a modified telegrapher equation.
As a consequence of this resemblance, it can be concluded that the shape transition from oscillating to monotonous conformational states provides furthermore evidence of a universal signature for a semiflexible polymer enclosed in compact space.
